Jordan Kyrou cut it to 4-1 at 12:34, scoring near the left post after he whiffed in front on a power play.
Tarasenko got St. Louis to within 4-2 at 4:14 of the third period with a one-timer from the right circle, but Kaprizov finished a 2-on-1 with Mats Zuccarello to push it to 5-2 at 11:47.
"I liked our push even throughout the whole game," Thomas said. "I thought 5-on-5 we played pretty well. I think we're pretty happy with (that). I mean, it felt like we were coming on, coming on, and a couple posts, a couple crossbars, and [Fleury] made some big saves over there, so you have to give him some credit.
"But I think we ought to be happy with our push and clean up a couple areas. We're happy with our game."
Eriksson Ek appeared to score his own hat trick at 12:04, but Berube challenged the play for offside, and the call was reversed after a video review.
Kaprizov then shot into an empty net during a 4-on-4 at 12:52 for a 6-2 final. It was the first playoff hat trick in Minnesota history.
"We talked after [Game 1 about] what he's going to have to do, or what are we going to have to do, to free [Kaprizov] up," Evason said. "We're not going to have to do anything. He's just going to do what he does. He's just going to play hard. Sure, he's got two, three people around him, they're trying to hit him. He's just going to play through it. He just stays the course, just goes about his business. It shouldn't surprise. … He takes it as a big challenge, and he did tonight and had a great game."